2....u is a unicity space for c1 (k , f.1) for all f.1 e g if and only if u satisfies property a....

TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 322, Number I, November 1990

L '-APPROXIMATION WITH CONSTRAINTS

ALLAN PINKUS AND HANS STRAUSS

ABSTRACT. In this paper we study problems of best L I-approximation to con-tinuous functions from finite-dimensional subspaces under a variety of con-straints. Included are problems of bounded coefficient approximation, approxi-mation with interpolation, restricted range approximation, and restricted range and derivative approximation. Emphasis is placed on problems of uniqueness.

1. INTRODUCTION

The literature of approximation theory abounds with numerous results on best approximation under a variety of constraints. These results are mainly concerned with such problems in the L 2 or uniform (L 00) norm. See, for example, the review articles by Chalmers [1] and Chalmers and Taylor [2], and the numerous references therein. In most of these problems, existence is easily established, and thus the questions considered are those of characterization, uniqueness, and algorithms for calculating best approximations.

Recently much progress has been made on the question of uniqueness of the best approximation in the L' -norm, from finite-dimensional subspaces, where both the subspace and the approximated functions are continuous. Similar progress has also been made on the analogous one-sided approximation problem. In this paper we consider problems of characterization and uniqueness for best L' -approximants to continuous functions, where the approximating sets are specified convex subsets of finite-dimensional subspaces. These convex subsets mayor may not depend on the function being approximated. In §3 we consider the problem of bounded coefficient approximation, while in §4 we deal with best approximation under interpolatory constraints. In §5 and §6 we discuss the problem of best restricted range approximation. §5 contains the more theoretical aspects of this problem, while in §6 we consider various examples. In §7 we show how these results can be generalized to restricted range and derivative approximation.

2. BACKGROUND

Let us first fix some notation. K will denote a compact subset of jRd sat-isfying K = int K. We let J1 be any nonatomic positive finite measure on K

Received by the editors November 23, 1988. 1980 Mathematics Subject Classification (1985 Revision). Primary 41A29; Secondary 41A52,

41A50.

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240 ALLAN PINKUS AND HANS STRAUSS

with the property that every real-valued f E C(K) (continuous functions on K ) is ,u-measurable, and such that if

IIflll := L If(x)1 d,u(x) = 0

for f E C(K) , then f = 0, i.e., 11·11 I is truly a norm on C(K). For notational ease we denote the set of such measures by Slf . We let CI (K,,u) denote the linear space C(K) equipped with norm 11·11 I' C I (K , ,u) is not a Banach space, i.e., it is not complete. However, it does have the important property that its dual space is L 00 (K , ,u) , i.e.,

(CI (K, ,u))* = Loo(K,,u).

U will always denote a finite-dimensional subspace of C(K), and M a closed convex subset of U. To every f E C(K) there always exists a best L I (K , ,u )-approximant from M. We first state two well-known criteria char-acterizing best approximants from M in C I (K , ,u). We recall the standard notation Z(f) = {x: f(x) = O}, and N(f) = K\Z(f).

Theorem 2.1. Let f E C(K). Then the following are equivalent. (i) u * is a best L I (K , ,u )-approximant to f from M.

(ii) For every u E M

f[sgn(f-u*)](u-u*)d,u:s f lu-u*ld,u. JK JzU-u')

(iii) There exists an hE LOO(K, ,u) satisfying ( 1 ) I h (x) I = 1 , all x E K , (2) fK hu* d,u ~ fK hu d,u, all u EM, (3) fKh(f - u*)d,u = Ilf - u*III'

The only nonstandard statement in the above theorem is (1) of (iii). It is only here that one uses the fact that our measure is nonatomic. With this fact, the result follows by using Liapounofi's Theorem. See Phelps [10]. The characterization (ii) is the usual one given. However, we will need the stronger statement (iii) in much of what follows (cf. §3).

The question of when M is a unicity set for CI (K, ,u), i.e., when to each f E C(K) there exists a unique best LI(K, ,u)-approximant from M, is a different and more difficult question. If M = U, then there are two known results characterizing unicity spaces. These are:

Theorem 2.2 (Cheney and Wulbert [3]). U is a unicity space for CI (K,,u) if and only if there does not exist an h E L 00 (K , ,u) and a u * E U, u * 1: 0, for which

( 1 ) I h (x) I = 1 , all x E K , (2) fK hu d,u = 0, all u E U, (3) hlu*IEC(K).


L I -APPROXIMA nON WITH CONSTRAINTS 241

Theorem 2.3 (Strauss [23]). U is a unicity space for C1 (K, fl) if and only if the zero function is not a best L 1 (K , fl )-approximant to any g E U*, g i= 0, where

U* = {g: g E C(K), Igl = lui for some u E U}. The only if statement of both these related theorems follows from the fact

that au* is a best LI(K, fl)-approximant to hlu*1 E U* for all a E [-1, 1], if h is as in Theorem 2.2.

If M = ~(f) := {u: u E U, u ::; f}, then it has been proved (see Strauss [24]) that to each f E C(K) there exists a unique best L 1 (K , fl )-approximant to f from ~ (f) (assuming ~ (f) =f. (2) ) if and only if for every u E U, u =f. ° , the zero function is not a best LI(K, fl)-approximantto lui from ~(Iul). Note the similarity to Theorem 2.3 in that the uniqueness question for all f E C(K) reduces to the uniqueness question on a set of "test" functions. There is also a different characterization if U contains a strictly positive function, in terms of quadrature formulae for U [12].

For arbitrary M as above, we have the recent result of Shi [18].

Theorem 2.4. Let M be a closed convex subset of U. Then M is a unicity set for C1 (K, fl) if and only if there does not exist agE C(K) and u1 ' u2 EM, u 1 =f. u2 ' such that

(a) (g - ul)(g - u2 ) = 0, (b) (u 1 + u2 )/2 is a best LI (K, fl)-approximant to g from M.

If M = U , then Theorem 2.3 is an immediate consequence of this theorem and is obtained by translating g by (u 1 + u2)/2.

It has been noted that the various necessary and sufficient conditions delin-eated for U or M to be a unicity set are fl dependent. That is, M may be a unicity set for C1 (K, fl) for some measure but not a unicity set for other mea-sures. As such, it is natural to ask for necessary and sufficient conditions on M implying that it is a unicity set for C1 (K, fl) for all "nice" measures fl. This problem in the case M = U has been considered by Kr06 [7], Sommer [19], and Pinkus [11]. The analogous problem for one-sided L 1 (K , fl )-approximation was considered by Pinkus and Totik [15] and Pinkus and Strauss [13].

To explain the result obtained in the case M = U, let us restrict our set of measures fl to

9 = {dfl = w(x) dx: WE C(K), W > ° on K}. For each u E U, u =f. 0, the (relatively) open set K\Z(u) (= N(u)) is the union of a possibly infinite, but necessarily countable number of open disjoint connected subsets of K, i.e., K\Z(u) = U:1 Ai' where the Ai are open, dis-joint, and connected. For convenience we also introduce the following notation. [K\Z(u)] will denote the number of open connected disjoint components of K\Z(u) , and for each u E U,

U(u) = {v: v E U, v = ° a.e. on Z(u)} ,



where the a.e. (almost everywhere) is with respect to Lebesgue measure. We say that U satisfies Property A if to each u E U, u -::j:. 0, with K\Z(u) = U:1 Ai' as above, and to each choice of ei E {-I, I}, i = 1, ... , m, there

exists a v E U(u), v -::j:. 0, satisfying eiv 2: ° on Ai' i = 1, ... , m. We can also state Property A in two other equivalent forms. Namely, U satisfies Property A if to each g E U* , g -::j:. 0, there exists a u E U, u -::j:. 0, satisfying u = ° a.e. on Z (g) , and ug 2: 0. Alternatively, U satisfies Property A if to each u E U, u -::j:. 0, and hE LOO(K) with

(a) Ih(x)1 = 1, all x E K , (b) hlul E C(K),

there exists a v E U, v -::j:. 0, satisfying hv 2: ° . The following result was proven with restrictions by Kroo [7] and Pinkus

[11], and in this form by Kroo [8]. Schmidt [17] later proved a somewhat more general result. Theorem 2.5. U is a unicity space for C1 (K , f.1) for all f.1 E g if and only if U satisfies Property A.

This result raises the natural question of which subspaces satisfy Property A. We know of two necessary conditions implied by Property A. To explain one of these conditions, we say that U decomposes if there exist nontrivial subspaces V and W of U such that U = V EEl W, i.e., U = V + Wand V n W = {a}, such that (K\Z(v)) n (K\Z(w)) = 0 for all v E V and w E W. In other words, there exist disjoint subsets Band C of K such that every function in V vanishes identically off B, while every function in W vanishes identically off C. With this definition, we can now state the following result. Theorem 2.6 (Pinkus and Wajnryb [16]). If U satisfies Property A, then

(1) [K\Z(u)]::; dim U(u), for all u E U. (2) If Z(U) = nUEUZ(u). and [K\Z(U)] 2: 2, then U decomposes.

If K c ~, then based on these results a full characterization of those U satisfying Property A may be given. From (2) of Theorem 2.6, it suffices to state this result for K = [a, b].

Theorem 2.7 (Pinkus [11], [12]). Let U be a finite-dimensional subspace of C[a, b]. Then the following are equivalent.

( 1) U satisfies Property A. (2) [[a, b]\Z(u)] ::; dim U(u) for all u E U. (3) (a) If [c, d], a < c < d < b, is a zero interval of u E U, then

(i) there exists a v E U for which

v(x) = { u(x), 0,

(ii) there exists aWE U for which

{ a, a<x<d, w(x) = -

u(x), d ::; x ::; b.


L 1 -APPROXIMA nON WITH CONSTRAINTS 243

(b) For a ::; c < d ::; b, let

~ d = {u: u E U, u = 0 on [a, c) U (d, b]).

Then each ~ , d is a W T -system (if ~, d i= {O}). Furthermore, if v E ~ d has no zero interval in [c, d], then v has at most dim ~, d-l

distinct zeros in (c, d) .

For K C ~d, d 2: 2, a full characterization is not yet known. In the case of one-sided L 1 (K , .u )-approximation, the answers to the analogous questions are somewhat different. Since we will need these results in §5, we review them here.

Let K and .u be as above. For each f E C(K) , set

V(f) = {u: u E U, u::; f}.

If V (f) i= 0 , then a best L 1 (K , .u )-approximant to f from V (f) necessarily exists. We will say that U is a unicity set for C1+(K,.u) if to each f E C(K) with V(f) i= 0, there exists a unique best LI(K, .u)-approximant to f from V(f) .

Theorem 2.8 (Pinkus and Totik [15]). U is a unicity set for C1+(K,.u) for all .u E 9 if and only if there exists a basis u1 ' •.. , un for U such that

(a) U i 2: 0, i = 1 , ... , n, (b) N(uJ n N(u) = 0, all i i= j.

This result says that the unicity set property holds in this one-sided case for all .u E 9 only if U has a very simple, trivial form. It is trivial in the sense that the approximation problem for U satisfying (b) reduces to n independent one-dimensional approximation problems.

The story, however, does not end here. If we restrict ourselves to continuously differentiable functions, then much more can be said if we add some additional minor assumptions. Before stating these assumptions, let us note that f E C I (K) , for K compact, if f has an extension to some open neighborhood of K , where f and its first partial derivatives are all continuous.

Assumption I. We assume that U c C I (K) and ( 1) U contains a strictly positive function, (2) K is a compact, convex subset of ~d , with piecewise smooth boundary

(and K = intK). For fECI (K) , we let ZI (f) denote the set of zeros of f on K for which

the following hold: (a) If x E int K , then all partial derivatives of f at x vanish. (b) If x E 8K, then all directional derivatives to f at x vanish for all

directions tangent to K at x.



If K is the interval [c, d), then Z, (f) is simply the set of zeros of f in [c, d), with the proviso that if x E (c, d) , then r (x) = o.

Finally we say that U c C' (K) satisfies Property B if to each u E U, u =I- 0, there exists a v E U, v =I- 0, satisfying

(1) Z, (u) ~ Z, (v), (2) v 2: o.

The following result was proven by Pinkus and Strauss (13].

Theorem 2.9. Let U c C' (K), and Assumption I hold. For every f1 E 9 and each f E C' (K) with ~ (f) =I- (2), there exists a unique best L' (K , f1)-approximant to f from ~ (f) if and only if U satisfies Property B.

3. BEST APPROXIMA nON WITH COEFFICIENT CONSTRAINTS

Let dim U = n, and u, ' ... , un be a basis for U. Given 0: = (a, ' ... , aJ and P = (P, ' ... , Pn ) satisfying -00 ~ a i < Pi ~ 00, i = 1, ... , n, set

The problem we consider in this section is that of best approximating f E C(K) from M(o:; P) in the L' (K, f1)-norm. This same problem in the uniform norm was studied by Pinkus and Strauss [14].

For u* = L7=, a>i E M(o:; P), set

{I,

b; = 0, -1,

a; = Pi' ai < a; < Pi'

* ai = ai'

and

From Theorem 2.1, we easily obtain

Theorem 3.1. Let f E C(K)\M(o:; P) and f1 E .s1/. Then the following are equivalent:

(i) u* = L7=, a;u i E M(o:; P) is a best L'(K, f1)-approximant to f from M(o:; P).

(ii) For all u E U(b*)

r [sgn(f - u*)]udf1 ~ r luldf1. } K }Z(f-u')


L I -APPROXIMATION WITH CONSTRAINTS 245

(iii) There exists an h E L 00 (K , J1) satisfying ( 1 ) I h (x) I = 1, all x E K ,

{ ~ 0, if b; = 1,

(2) 1 h U j d J1 = 0, if b; = 0, K ~o, zjb;=-1,

(3) 1 h(f - u*) dJ1 = Ilf - u*III'

Statement (iii) provides us with the more precise information needed for the solution of the unicity problem. For this unicity problem, we also need an important addendum to (2) and (3) of (iii).

Lemma 3.2. Let f, u*, and h be as in Theorem 3.1 (iii). If u = 2:7=1 ilju j is any other best LI(K, J1)-approximant to f from M(o.; ft), then

(a) IK h(f - u) dJ1 = Ilf - ull l ,

(b) if IKhujdJ1 =I- 0 for some j E {1, ... , n}, then

il = a * = { P j , if b; = 1 , } } ;I'b* 1 Ct j , ZJ j = - .

Proof. From (1), (2), and (3) of (iii) of Theorem 3.1,

Ilf - u*111 = 1 h(f - u*)dJ1 ~ 1 h(f - u)dJ1 ~ IIf - ull l ·

Since equality holds, we immediately obtain (a) and

1 h(u* - u)dJ1 = O.

That is,

From (2) of (iii) of Theorem 3.1, we have that

(a; -aj) 1 hU j dJ1 ~ 0

if Ct j ~ aj ~ Pj and i E {1 , ... , n}. Thus

(a; - 0) 1 hU j dJ1 = 0, i = 1, ... , n.

If IK hUj dJ1 =I- 0, it follows that ilj = a; and (b) holds as a consequence of (2) of (iii) of Theorem 3.1. 0

The seemingly innocuous Lemma 3.2 has a simple and important conse-quence. Set



and J = {I, ... , n}V . Every best L 1 (K, .u)-approximant to f from M(a; p) of the form 2:7=1 iiju j satisfies iij = a; for all j E J, Let

7=f-2: a;Uj' JEJ

Then it = 2:7=1 iiju j is a best L 1 (K , .u)-approximant to f from M(a; P) if and only if 2:jEIiijuj is a best LI(K, .u)-approximant to 7 from MI(a; P), where

Let

MI(a; P) = {2: aju j: a j S aj S Pj' i E I} . lEI

VI = span{u j : i E I}. From Theorem 3.1 (iii) and Lemma 3.2, we have

(1) Ih(x)l= 1 all xEK, (2) hhujd.u = 0, all i E I, (3)

(a) 1 h (7 - 2:a~uj) d.u = 117 - 2:a~ujll ' K lEI lEI 1

(b) Ix h (7 - ~iliUi) d~ ~ 117 - p,ul From Theorem 2.1 (iii), this implies that both 2: jEI < uj and 2:iEI iiiUj are

1 -best L (K, .u)-approximants to f from VI' and not only from MI(a; P) . We have therefore proved:

Theorem 3.3. Let N = {i: a i = -00, Pj = oo}. If span{u j , ••• , uj } is a , k

unicity space for C1 (K, .u) for every choice of distinct {ii' ... , ik } satisfying N <;; {ii' ... , ik } <;; {I, ... , n}, then M(a; P) is a unicity set for C1 (K, .u).

It is not clear that the converse result is valid in this generality. It is, however, true with a minor additional assumption.

Theorem 3.4. Let N = {i: a j = -00, Pi = oo}. Assume that for all i tI. N, we have -00 < ai < Pi < 00. If M(a; P) is a unicity set for C1 (K, .u), then span {u , ... , U } is a unicity space for C1 (K , .u) for every choice of distinct I, Ik

{ii' ... , ik } satisfying N <;; {ii' ... , ik } <;; {I, ... , n}. Proof. Let

N <;; {i l , ... , ik } <;; {I, ... , n}, and assume that Vk = span{u i , ' ... , u jk } is not a unicity space for C1 (K, .u). By definition there exists an f E C(K) and u l , u2 E Vk , u l =1= u2 , such that both uland u2 are best L 1 (K , .u )-approximants to f from Vk . Since the


L1-APPROXIMATION WITH CONSTRAINTS 247

set of best approximants is convex and u' + u, r = 1 , 2, are best L 1 (K , f-l)-approximants to f + U from Uk for each U E Uk ' we may assume that

k ] L] U = a· u· , I j IJ j=]

where a i < a; < Pi' r = 1,2; j = 1, ... , k. From Theorem 2.1(iii), there J J J

exists an h E L 00 (K , f-l) satisfying (1) Ih(x)l= l,all xEK, (2) fK hUi df-l = 0, j = I, ... , k, (3) fKh(f - u')df-l = Ilf - u'II], r = 1,2.

Set Q = {I, ... , n}\{i] , ... , i k }. For each i E Q we have -00 < a i < Pi < 00. Let i E Q. If fKhuidf-l = ° let Yi E [ai' PJ. If fKhuidf-l > 0 set Yi = Pi' while if fK hUi df-l < 0 set Yi = a i . Define

it = LYiU i iEQ

and J = f + it, it' = u' + it, r = 1,2. From Theorem 3. 1 (iii), we have that both it] and it2 are best L] (K , f-l )-approximants to J from M (0:; P). 0

We can now put together Theorems 2.5, 3.3, and 3.4 to obtain the analo-gous necessary and sufficient conditions for when M(o:; P) is a unicity set in C](K, f-l) for all f-l E 9. Since Chebyshev (T-)systems satisfy Property A on [a, b], we obtain, for example, that if {u i ' ••• , ui } is a T-system on

1 k

[a, b] for every choice of {i], ... , i k } ~ {I, ... , n} and k = 1, ... , n, then M(o:; P) is a unicity set for C] (K , f-l) for all f-l E 9, and for any choice of 0: and P as originally defined. A sequence of functions {u]' ... , un} , such that every subsequence is a T-system, is called a Descartes system.

4. BEST APPROXIMATION UNDER INTERPOLATORY CONSTRAINTS

Assume that dim U = n, and that we are given m distinct points t] , ... , tm in K. We wish to approximate f E C(K) from the set of U E U satisfying u(t]) = f(ti) , i = 1, ... , m. For this problem to be well defined, we assume that this interpolation is always possible. That is, we assume that U is of dimension mover {t], ... , t m}. Equivalently, we can restate this condition as follows. Set

v = {u: U E U, u(ti ) = 0, i = 1, ... , m}.

Then V is a subspace of U of dimension n - m. If m = n, then these interpolating conditions uniquely determine the admissible u, and there is no approximation involved. As such, we always assume in what follows that 1 ::; m <no

For notational ease, set

M(f) = {u: U E U, u(ti) = f(t), i = I, ... , m}.



Our problem is to approximate f from M(f). Obviously M(f) depends on f. Nonetheless M(f) is a closed convex subset of U, and as such, a best L 1 (K , ,u )-approximant necessarily exists. It is important to note that if U E M(f), then

M(f) = {u+v: v E V}.

Using this form of M(f) we may apply Theorem 2.1 to obtain a characteriza-tion of best Ll(K, ,u)-approximants from M(f).

Theorem 4.1. Let f E C(K). Under the previous assumptions, the following are equivalent:

(a) u* is a best Ll(K, ,u)-approximant to f from M(f). (b) For every v E V

I r sgn(f - u*)v d,ul:s r Ivl d,u. h I }Zu-u*)

(c) There exists an h E L 00 (K , ,u) satisfying ( 1 ) I hi = 1, all x E K , (2) JK hv d,u = 0, all v E V, (3) JK h(f - u*) d,u = Ilf - u*lll'

We also have the following result.

Theorem 4.2. To each f E C( K), there exists a unique best L 1 (K , ,u )-approxi-mant from M(f) if and only if V is a unicity space for C1 (K , ,u). Proof. (¢:) Let f E C(K) and U E M(f). Since V is a unicity space for C1 (K, ,u), there exists a unique best Ll (K, ,u)-approximant v* E V to f - U. The function u* = U + v* is necessarily the unique best L I (K , ,u)-approximant to f from M(f).

(=;.) Assume V is not a unicity space for C1(K,,u). From Theorem 2.2 there exists an hELoo(K,,u) and a V*EV, v*-:j:.O,suchthat hlv*IEC(K). From Theorem 2.1, it may be easily checked that aV * is a best L I (K , ,u)-approximant to f=hlv*1 from V,forevery aE[-I, 1]. Since v*(t)=O, i = 1, ... , m, we also have f(t) = 0, i = 1, ... , m, and therefore M(f) = V. We have constructed an f E C(K) with more than one best L 1 (K , ,u)-approximant from M(f).

Remark 4.1. The latter half of the proof of Theorem 4.2 epitomizes one of the principles implied by Theorems 2.2 and 2.3. Namely, to check the unicity space property of a finite-dimensional subspace V of C(K), it suffices to check it only on a set of "test" functions. Moreover, these "test" functions f necessarily satisfy f(x) = 0 if v(x) = 0 for all v E V. Since v(t) = 0, i = 1, ... , m, for all v E V, it is sufficient to verify the unicity space property for those f E C(K) satisfying f(ti) = 0, i = 1, ... , m, i.e., f E C(K) for which M(f) = V.


L 1 -APPROXIMA nON WITH CONSTRAINTS 249

Putting together Theorems 4.2 and 2.5, we obtain

Theorem 4.3. For each 11 E 9 there exists a unique best L1(K, Il)-approximant to every f E C(K) from M(f) if and only if V satisfies Property A.

The consequence of this result is generally negative in character. For example, if K = [a, b] and at least one of the points {tJ;:1 lies in (a, b), then either [K\Z(V)] 2:: 2 or all functions in V have their support between two consecutive t/s. If the former occurs, then it follows from Theorem 2.6 that V does not satisfy Property A unless V decomposes. Thus if U is a T-system on (a, b), at least one of the points {ti};:1 lies in (a, b), and 1 :::; m < n, then V does not decompose and therefore does not satisfy Property A. There thus exist a 11 E 9 and an f E C(K) without a unique best L I (K , 11 )-approximant from M(f).

This result is in marked contrast to the corresponding problem in the uniform norm (see Paszkowski [9]), where it is shown that uniqueness of the best uniform approximation to f from M(f) always holds if U is aT-system.

5. BEST RESTRICTED RANGE APPROXIMATION: THEORY

We assume that we are given two functions b, t E C(K) which satisfy b(x) < t(x) for all x E K. Set

M(b, t) = {u: u E U, b:::; u:::; t}.

The problem now under consideration is that of finding best L I (K , 11 )-approxi-mants to f E C(K) from M(b, t). One characterization of the best L I (K, 11)-approximants is given by Theorem 2.1 with M = M(b, t). However, this result is insufficient for our purposes. We first give a more particular characterization theorem, which is a consequence of Theorem 2.1(iii). To this end we will make the following assumption without which the next result is not valid. We assume there exists a U E U satisfying b(x) < u(x) < t(x) for all x E K. We write this as int M(b, t) =I- 0. Our characterization theorem generalizes results of Duffin and Karlovitz [4] and Gehner [5].

Theorem 5.1. Let M(b, t) be as just described and dim U = n. Then u* is a best L I (K, Il)-approximant to f E C(K) from M(b, t) if and only if there exist distinct points XI"'" Xm E Z(u* - b) and YI' ... 'Yr E Z(t - u*), 0:::; m + r :::; n, positive numbers {aJ;:1 and {Pi};=I' and an hE Loo(K, 11) satisfying

( 1) I h (x) I = 1 , all x E K , (2) fK hu dll + 2:;:1 aiu(xi) -2:;=1 Piu(y) = 0, all u E U, (3) fKh(f-u*)dll=llf-u*III'

Proof. (¢::) Assume (1), (2), and (3) hold. From the definition of the {XJ;:I' {YY=I' {aJ;:1 and {PJ;=I ' it follows from (2) that

(2') fK hu* dll 2:: fK hu dll, all u E M(b, t).



Theorem 2.1(iii), and (1), (2'), and (3) imply that u* is a best L1(K,f.l)-approximant to f from M(b, t).

(~) Assume u* is a best LI(K, f.l)-approximantto fE C(K) from M(b, t). Since M(b, t) is a closed convex set, we have from Theorem 2.1(iii) the exis-tence of an hE Loo(K, f.l) satisfying (1), (3), and

(2/1) IKhu* df.l ~ IKhudf.l, all u E M(b, t). Our aim is to prove (2) of the statement of the theorem. To this end, let u1' ... ,un be any basis for U, and set

P = {(u 1 (x), ... , un(x): x E Z(t - u*)}

U{(-u1(x), ... , -un(X)):XEZ(U* -b)}.

Let Q denote the closed convex cone generated by P. (If P = 0 , set Q = {O} .) Furthermore, set

c = (1 hU I df.l, ... , 1 hUn df.l) .

If c 1:. Q there exists a hyperplane strictly separating c from Q. Since Q is a cone, we may assume that this hyperplane passes through the origin. Thus there exists an a = (ai' ... , an) =I- 0 for which

taj 1 hujdf.l > 0 ~ tajqj j=1 j=1

for all q = (ql ' ... , qn) E Q. Set u = E;=I ajuj . Then IK hu df.l > 0, while u(x) ::; 0 for all x E Z(t-u*)

and u(x) ~ 0 for all x E Z(u* - b). Recall that it E U satisfies b < it < t. Therefore, for some f5 > 0, f5

sufficiently small, v = u - f5(u* - it) satisfies v < 0 on Z(t - u*), v > 0 on Z(u* - b), and IK hv df.l > O. A standard compactness argument implies that u* + ev E M(b, t) for e > 0, e sufficiently small. But IK h(u* + ev) df.l > IK hu* df.l contradicting (2/1). Thus CEQ.

Since CEQ, there exists {Xi};:I' {Yi};=I' {ai};:I' and {PJ;=I as in the statement of the theorem with 0::; m + r ::; n satisfying (2). 0

As a corollary to this result we have the more easily stated, but less useful

Corollary 5.2. Let M(b, t) be as described earlier and dim U = n. Then a best LI(K, f.l)-approximant to f E C(K) from M(b, t) if and only if

(5.1 ) r [sgn (f - u *)] u d f.l::; r I u I d f.l } K }z(f-u*)

for all u E U satisfying u::; 0 on Z(t - u*) and u ~ 0 on Z(u* - b).

* . U IS

Proof. If u* is a best L 1 (K , f.l)-approximant to f E C(K) from M(b, t) then (5.1) follows from (1), (2), and (3) of Theorem 5.1. If (5.1) holds, then we use Theorem 2.1 (ii) to obtain our result. 0


L I-APPROXIMATION WITH CONSTRAINTS 25'

Given 11 E S>f , let us now ask when M(b, t) is a unicity space for C, (K, 11) . As above, we assume that intM(b, t) =I- 0. We can now state the following analog of Theorem 2.2.

Theorem 5.3. Let M(b, t) be as above and dim U = n. M(b, t) is a unicity set for C, (K ,11) if and only if there do not exist U,' U2 E M(b, t), u, =I- u2' and

(a) {Xi}:' <:;:: Z(u, - b) n Z(u2 - b), (b) {yJ;=, <:;:: Z(t - u,) n Z(t - U 2 ),

( c) a i > 0, i = 1 , ... , m, and Pi > 0, i = 1 , ... , r, where 0 ::; m + r ::; n, and an h E L 00 (K , 11) for which

(1) Ih(x)1 = 1, all x E K, (2) fK hu dll + 2::, aiu(xi) - 2:;=, Piu(y) = 0, all U E U, (3) hlu, - u21 E C(K).

Proof. (~) Assume there exist u,' u2 E M(b, t) with {Xi}:" {y);=, , {a i }:" {PJ;=" and h satisfying (a)-(c) and (1)-(3). Set

f( ) = h(x)lu, (x) - u2(x)1 u, (x) + u2(x) x 2 + 2 .

Then f E C(K). Furthermore, as is easily checked,

i = 1,2,

on all of K. Thus

(3') fKh(f-ui)dll=llf-uill" i=1,2. Using (1), (2), and (3'), we obtain from Theorem 5.1 that both u, and u2

are best L' (K , Il)-approximants to f from M(b, t). Thus M(b, t) is not a unicity set for C, (K , 11) .

(¢::) Assume M(b, t) is not a unicity set for C,(K,Il). There exists an f E C(K) and u,' u2 E M(b, t), u, =I- u2 ' such that both u, and u2 are best L'(K, Il)-approximants to f from M(b, t). Since (u, + u2)/2 is also a best L' (K , Il)-approximant to f from M(b, t) , we easily see that

21 (f - (U, ; U2)) (X)I = l(f - u,)(x)1 + l(f - u2)(x)1

for all x E K. Thus

Z (f- U, ;U2) <:;::z(f-u,)nz(f-u2).

Because (u, + u2 )/2 is a best L' (K, Il)-approximant to f from M(b, t) we have {xJ:" {Yi };=" {a i}:" {Pi };=, and h satisfying the various condi-tions given in Theorem 5.1 with respect to (u, + u2)/2. Thus, (1) and (2) of this theorem hold. Since u,' u2 E M(b, t) we immediately obtain (a)-(c). It remains to prove (3) of this theorem.



From Theorem 5.1, we have

l h (f - U I ; U2 ) dp = 1\1 - ul ; u2111 . Thus, using (1) we have h = sgn(J - (u l + u2 )/2)p a.e. on N(J - (u l + u2 )/2). We alter h on a set of p-measure zero so that h = sgn(J - (u l + u2)/2) on all of N(f - (u l + u2 )/2) , while maintaining (1) and (2). Consider hlu l - u21. If hlu l - u21 is discontinuous at a point x E K, then x is necessarily a point of discontinuity of h. Every point of discontinuity of h occurs, by construction, at a zero of f - (u l + u2)/2. But

Z(J - (u l + u2)/2) ~ Z(f - uI) n Z(J - u2) ~ Z(u l - u2) = Z(Ju l - u2J), and lUI - u21 is continuous. Thus, hlu l - u21 is continuous at each point of discontinuity of h, implying that hlu l - u21 E C(K). 0

The characterization of a unicity set as determined by Theorem 5.3 is gen-erally difficult if not impossible to verify. An easier sufficient condition is the following:

Proposition 5.4. Let M(b, t) be as shown before and dim U = n. If for each g E U*, g =I- 0, and every set of {XJ::I' {Yi};=1 ~ Z(g) with 0:::; m + r :::; n, there exists a U E U satisfying u(x) ~ 0, i = 1, ... , m; u(y):::; 0, I, ... , r, and

(5.2) { [sgng]udp > ( luldp, lK lZ(g)

then M(b, t) is a unicity set for CI (K, p). Proof. If M(b, t) is not a unicity set for CI (K, p), then we have the existence of {X)::I' {Yi};=I' {ai}::I' {Pi};=I' h, u l and u2 satisfying (a)-(c) and (1)-(3) of Theorem 5.3. From (1) and (3), g = hlu l - u21 E U*, g =I- 0, and h = sgn g on N(g). By construction {XJ::I' {YY=I ~ Z(g). Let u* E U satisfy (5.2) with respect to this g and these {XJ::I and {Yi};=1 . From (2)

1 hu* dp = - faiu*(Xi) + tpiU*(Yi):::; o. K i=1 i=1

Thus { [sgng]u* dp + ( hu* dp:::; 0

lK lZ(g) which implies that

{ [sgn g] u * d p :::; - { h u * d p:::; ( I u * I d p . 1 K lZ(g) lZ(g)

This contradicts the fact that u* satisfies (5.2). 0

The conditions of Theorem 5.3 and Proposition 5.4 very much depend on the choice of p E.sf . We wish to obtain conditions on U and M(b, t) that would imply that M(b, t) is a unicity set for C I (K, p) for all PEg. To this end, we first introduce a condition we call Property C. As always, int M(b, t) =I- 0.


L I -APPROXIMATION WITH CONSTRAINTS 253

Definition 5.1. We say that M(b, t) satisfies Property C (with respect to band t), if for any ul ' u2 E M(b, t), u l 1:- u2 ' with {Xi}:l ~ Z(u l -b)nZ(u2 -b), {yJ;=l ~z(t-ul)nz(t-u2)' OSm+rSn, K\Z(u l -u2)=U;=lAi ,and every choice of e i E {-I, I}, i = 1, ... , k, there exists a v E U, v 1:- 0 satisfying

(1) v = 0 a.e. on Z(u l - u2),

(2) eiV 2: 0 on Ai' i = 1, ... , k, (3) v(x) 2: 0, i = 1, ... , m; V(Yi) SO, i = 1, ... , r.

The a.e. of (1) is with respect to Lebesgue measure. There is a simple equiv-alent definition of Property C that we will also use.

Definition 5.2. We say that M(b, t) satisfies Property C (with respect to band t) if for any ul ' u2 E M(b, t), ul 1:- u2 ' with {XJ:l ~ Z(u l -b)nZ(u2 - b), and {Yi};=l ~ Z(t - u l ) n Z(t - u2 ), 0 S m + r S n, and any h E Loo(K) satisfying

(a) Ih(x)1 = 1, all x E K, (b) hlu l - u21 E C(K) ,

there exists a v E U, v 1:- 0, for which (1) v = 0 a.e. on Z(u l - u2 ),

(2) hv 2: 0 a.e., ( 3 ) v (x i) 2: 0, i = 1 , ... , m; v (y i) SO, i = 1 , .. . , r .

Condition (1) of Definition 5.2 is actually a consequence of the other condi-tions; see the proof of Theorem 5.5. The importance of Property C is a result of this next theorem.

Theorem 5.5. M(b, t) is a unicity set for Cl (K, f-l) for every f-l E 9 if and only if M(b, t) satisfies Property C (with respect to band t). Proof. ('*) Assume Property C does not hold. As such, using Definition 5.2, there exist u l ' u2 E M(b, t), u l 1:- u2 ' points {X):l ~ Z(u l - b) n Z(u2 - b) and {Yi};=l ~ Z(t - u l ) n Z(t - u2), 0 S m + r S n, as well as an hE Loo(K) satisfying (a) and (b) of Definition 5.2, and such that no v E U, v 1:- 0, satisfies (1)-(3) therein.

We will prove the existence of ji E 9, iii 2: 0, i = 1, ... , m, Pi 2: 0, i = 1, ... , r, and an h E L 00 (K ,ji) (not quite the previous h) such that (a)-(c) and (1)-(3) of Theorem 5.3 are satisfied.

To this end, set

f; = {u: U E U, U = 0 a.e. on Z (u 1 - u2 )} ,

l.e., f; = U(u l - u2). f; is a subspace of U of dimension s, where 1 S s S n. Let

f; = span{v l , ... ,vJ and W = span{vs+ l , ... , vn }



where the {VJ;=1 span U. Note that the vS+ l ' ..• ,vn are linearly independent on Z(u l - u2 ). We alter h on Z(u l - u2 ) by setting it equal to h thereon, where h is defined by

(1) Ih(x)1 = 1, all x E Z(u l - u2),

(2) JZ(U,-U2) hw dx = 0, all WE W.

Such an h necessarily exists. Note that this new h still satisfies (a) and (b) of Definition 5.2, and has no effect on (1)-(3) thereof.

Let B ~ ~n be defined by

B ~ { (l hVjd~ + ~aiVj(Xi) - ~ PiVj(Y,)r~1 : ~ E!if, ai' Pi ;, o} . Since 9 is a convex cone, the set B is a convex cone. If 0 E f) B , then there exists an a = (ai' ... , an) f=- 0 that defines a support hyperplane to B at O. Set v = I:]=l ajvj . Then v f=- 0 and

for all fl E 9 and ai' Pi ~ O. This implies that hv ~ 0 a.e. on K, v(xi ) ~ 0, i = 1, ... , m, and v(y) :::; 0, i = 1, ... , r. Now v = U + w where U E f) and w E W. Furthermore, hv = hw on Z(u l - u2). Thus hw ~ 0 a.e. on Z(u l - u2). This contradicts (2) in the definition of h unless w = O. Thus v = U E f). That is, v = 0 a.e. on Z(u l -u2 ) and hv ~ 0 a.e. This contradicts our assumption that Property C does not hold. Thus 0 tt. f) B .

Since B is a convex cone, we have 0 E B. Thus there exists a jl E 9 and ai' Pi ~ 0 such that

for all U E U. From Theorem 5.3 it follows that M(b, t) is not a unicity set for Cl(K, jl).

(¢::) Assume Property C holds. If M( b, t) is not a unicity set for C l (K , fl) , some fl E 9 , we have the existence of u l ' u2 ' {xJ, {yJ, {aJ, {Pi}' and h satisfying (a)-(c) and (1)-(3) of Theorem 5.3. Let v E U, v f=- 0, satisfy (1)-(3) of Definition 5.2 with respect to these terms. This immediately contradicts (2) of Theorem 5.3. Thus M(b, t) is a unicity set for C l (K ,fl) for every flE9. 0

While Property C is the correct characterization, it is also generally difficult to verify for a specific band t. However, we can totally characterize those U for which M(b, t) satisfies Property C for all admissible band t.



Theorem 5.6. Assume dim U = n. Then M(b, t) satisfies Property C for all b, t E C(K) for which int M(b, t) -I- 0 if and only if there exists a basis v, ' ... , Vn for U such that

(1) N( vJ is connected for each i, (2) N(vi) n N(v) = 0 for all i -I- j.

Proof. (~) Assume M(b, t) satisfies Property C for all b, t E C(K) for which intM(b, t) -I- 0. Then there must exist a unique best approximant to t from W(t) = {u: u E U, u:::; t}, for every t E C(K). From Theorem 2.8 this implies that there exists a basis v,, ... ,vn for U satisfying (2). Since there must also exist a unique best approximant to every f E C(K) from all of U, it follows from Theorem 2.6 that [K\Z(u)]:::; dim U(u) for all u E U. Let u be any of the basis elements v,, ... , vn . Since dim U(u) = 1, we obtain (1).

(~) If U has a basis v,, ... ,vn satisfying (2), then it is easily see that M(b, t) is a unicity set for C, (K, f.l) if and only if

Mi(b, t) = {u: u E V;, u E M(b, t)}

is a unicity set for C, (K , f.l) ,where V; = span{ Vi}' i = 1, ... , n. Conditions ( 1 ), (2), and (3) of Definition 5.1 are easily verified in this case. For if U,' u2 E MJb, t), u, -I- u2 ' then since Z(u, -b)nZ(u2 -b) = Z(u, -u2 ) = Z(vi) , we have v = 0 on Z(u, - u2) for all v E V;, with the similar result for Z(t-u,)nz(t-u2 ). Thus, (3) holds. Furthermore, from (1), K\Z(u, -u2 ) = N( vJ and this is a connected set. Thus, (1) and (2) of Definition 5.1 are obtained by setting the v therein to be Vi or -Vi' D

The subspaces satisfying the conditions of Theorem 5.6 are uninteresting as in the case of one-sided approximation. However, paralleling Theorems 2.8 and 2.9, we should also consider the case where differentiability is present. Using the arguments of Theorem 5.6, we can prove:

Proposition 5.7. Let U c C' (K) and Assumption I hold. If M(b, t) satisfies Property Cfor all b,t E C'(K) with intM(b, t) -I- 0, then U must satisfy Property A and Property B.

See §2 for the definitions of Assumptions I and Properties A and B. It is an open question as to whether Assumption I and Properties A and B imply Property C.

6. BEST RESTRICTED RANGE APPROXIMATION: EXAMPLES

In this section, we give examples of M(b, t) satisfying Property C. We will restrict ourselves to the case of an interval, i.e., K = [c, d), and assume that UcC'(K), b, tEC'(K) and intM(b,t)-I-0.

We first give a sufficient condition and then delineate some subspaces satisfy-ing this condition. For ease of notation, we introduce the following terminology.



Definition 6.1. For any set of points W ~ [c, d], we let IWI denote the number of points in W, and /(W) denote the number of points in W where every point in W n (c, d) is counted twice.

Definition 6.2. Let fEet [c, d]. Then Z* (f) counts the number of zeros of f in [c, d], where a zero x E (c, d) is counted twice if f(x) = I (x) = O. The analogous definitions hold for the open and half-open intervals. Similarly Z (f) counts the number of zeros in the same way as Z*(f) , but zeros at endpoints may be counted twice if both the function and its derivative vanish there.

Thus for any f E Ct[c, d], Z*(f) :::; Z(f). For a given U c C[c, d], and any c :::; e < f :::; d , we also define

v:,,f = {u: U E U , U = 0 on [c, e) U (f, dJ) .

For notational ease, we also set

We can now state:

{

[e, f]'

[[e, fJ] = [e, f) , (e, fJ, (e, f),

if c = e f = d , , if c = e, f < d, if c < e, f = d, if c < e, f < d.

Theorem 6.1. Let U c C t [c, d], dim U = n. Assume U satisfies Property A, and if U E v:,,f has no zero intervalin [e, fJ, then

(6.1) Z*(ul[[e,fll) :::; dim v:,,f - 1.

Then M(b, t) satisfiesPropertyCforall b, t E Ct[c, d] with intM(b, t) =f. 0. Proof. We are given b, t E Ct[c, d] with intM(b, t) =f. 0. Let ut ' u2 E

M(b, t), ut =f. u2 · Assume {xi};:t ~ Z(u t - b) n Z(u2 - b), {Yi};=t ~ z(t-u t )nZ(t-u2) where O:::;m+r:::;n,and hEL=(K) satisfies

(a) Ih(x)1 = 1, all x E [c, d], (b) hlu t - u21 E C[c, d].

We will construct a v E U, v =f. 0, satisfying (1)-(3) of Definition 5.2. Suppose that [e, fJ is a maximal interval on which u t - u2 has no zero

interval. That is, ut - u2 has no zero interval on [e, fJ, if c < e then ut - u2 vanishes identically on (e - e, e) for some e > 0, and if f < d then u t - u2 vanishes identically on (f, f + e) for some e > O. Such an [e, fJ exists as a consequence of Property A. Furthermore from Theorem 2.7 (3), there exists a W E v:,,f such that W = u t - u2 on [e, fJ .

Let us assume, for convenience, that c < e < f < d, i.e., [[e, f]] = (e, f) . Set

R = {{x):t U {y);=t} n (e, f). Since b, t E Ct[c, d], we have w(x) = w'(x) = 0 for all x E R. let

S = {x: x E (e, f), w(x) = 0, x t!. R}.


L'-APPROXIMATION WITH CONSTRAINTS

By assumption I(R) + lSI:::; dim~,f - 1.

For e > 0, e small, set

Then

T£ = {x - e , x + e: x E R, h does not change sign at x} U {x: x E R uS, h changes sign at x} .

1= IT£I :::; I(R) + lSI:::; dim v:,,f - 1.

257

Since v:,,f is a weak Chebyshev (WT-) system (see Theorem 2.7 (3)) it follows from Sommer and Strauss [20] and Stockenberg [21] that v:,,f contains an (l + I)-dimensional WT-system V. From a theorem of Jones and Karlovitz [6], there exists a v£ E V ~ r::,,f' Ilv£lloo = 1, which changes sign weakly at each of the points of T£. Let v be a limit point of v £ as e 1 O. By construction v =j:. 0, Jhv ~ 0 for some J E {-I, I}, and v(x) = 0 for all x E R. Set v = Jv. Then v satisfies (1 )-(3) of Definition 5.2, i.e., Property C holds.

The other cases, i.e., e = e and/or f = d, are similarly treated. 0

Corollary 6.2. Let U c e'[e, d], dim U = n. If

(6.2) Z*(u) :::; n - 1

for all u E U, u =j:. 0, then M(b, t) satisfies Property C for all b, tEd [c, d] with intM(b, t) =j:. 0.

A subspace U c e' [c, d] of dimension n is said to be an extended Cheby-shev system of order 2 (ET2-system) if Z(u) :::; n - 1 for all u E U, u =j:. O. Since Z * (f) :::; Z (f) for all fEe I [e , d], we immediately obtain that if U is an ET2-system, then it satisfies (6.2). Polynomials are examples of ET2-

systems. We now give two additional examples of subspaces satisfying the conditions

of Theorem 6.1.

Example 6.1. Polynomial splines with simple, fixed knots. Let n ~ 3 and ~o = e < ~l < ... < ~k < d = ~k+l • Set

S = {s: s(x) = ~ aixi + t, bi(x - ~):-' , ai' bi E lR} . S is the space of polynomial splines of degree n - 1 with the simple, fixed knots {c;J ~=l . Since n ~ 3, we have See' [e, d]. Pinkus [12] proved that S satisfies Property A. It is also well known that if s E v:" f has no zero interval in [e, f] , then e = ~i and f = ~j , and

Z*(sl[[~,,~;ll):::; dim ~"~J - 1.

Thus, S satisfies the conditions of Theorem 6.1.



Example 6.2. Piecing together ET2-systems. Let c = eo < ... < ek < ek + 1 = d. On each interval Ii = [ei_ l , ei ], i = 1, ... , k + 1, let Vi C d(Ii ) , dim Vi = ni , ni ~ 2. Assume that the constant function is in Vi' and

V; = {u': u E V}

is a T-system on Ii' (Thus, Vi is an ET2-system on Ii') Let V c d[c, d] be the subspace defined by VII = Vi' i = 1, ... , k + 1. We claim that V satisfies the conditions of Theor~m 6.1.

From [12], we have that V satisfies Property A. It remains to prove (6.1). If u E ~,f has no zero interval in [e, f], then e = ei and f = ej for some o :S i < j :S k + 1 . It therefore suffices to prove (6.1) only for such e and f. Set Ve. e = V ., O:S i < j :S k + 1 . From [12], we have

I' j 1, }

(i) dim va,k+1 = n l + ... + nk+1 - 2k . (ii) For 1 :S j:S k, dim va,j = n l + ... + nj - 2j.

(iii) For 1 :S i :S k, dim ~, k+1 = ni+1 + ... + nk+1 - 2(k + 1 - i) . (iv) For 1 :S i < j:S k, dim ~,j = ni+1 + ... + nj - 2(j - i + 1).

If any of these values is negative, then we understand it to mean that dim V . = l,j

O. Similarly, if

~',j = {u': u E V, u' = 0 on [c, e) U (ej , d])

then ~/, j is a W T -system and

(i) dim V;,k+1 = n l + ... + nk+1 - 2k - 1. (ii) For 1 "5:. j "5:. k, dim V;,j = n l + ... + nj - 2j .

(iii) For 1 "5:. i"5:. k, dim ~',k+1 = ni+1 + ... + nk+1 - 2(k + 1 - i). (iv) For 1 "5:. i < j "5:. k, dim V;',j = ni+1 + ... + nj - 2(j - i + 1) + 1.

These facts are also contained in [12]. Note that the relationship between dim V . and dim V' . is not the same for all i and j.

I, j I, j

In what follows we will need a result of Stockenberg [22]. Namely, assume V c C[a, b] is a WT-system of dimension n, and for every x E (a, b) there exists a u E V satisfying u(x) =I- O. Then if u* E V has no zero interval in (a, b), we have that u* has at most n - 1 distinct zeros in (a, b).

With these results we can proceed to prove (6.1).

Case 1. i = 0, j = k + 1. u has no zero interval in [c, d]. Let Z* (u) = m on [c, d]. If u' has no zero interval in [c, d] , then u' Eva, k+ I ' and u' has at least m - 1 distinct zeros in (c, d). Using the above result of Stockenberg, applied to u', we get

m - 1 "5:. dim V~,k+1 - 1 = dim VO,k+1 - 2.

Thus m = Z* (u) "5:. dim va, k+1 - 1 . If u' has zero intervals in [c, d] , then the inequality is even sharper. There is simply a bit more bookkeeping involved.



Case 2. i = 0, 1 :::; j :S k. u E Va,} and u has no zero interval in [eo' e). Let z*(ul[eo,e) = m. Assume u' has no zero interval in [eo, e). Since u(e) = u' (e) = 0, u' has at least m distinct zeros in [eo' e) . Thus

m :::; dim V~,} - 1 = dim Va,} - 1 ,

and Z*(ul[eo,e):::; dim Va,} - 1. Again if u' has zero intervals in [eo' e) the result also holds.

The remaining two cases are similarly proven. 0

7. BEST RESTRICTED RANGE AND DERIVATIVE APPROXIMATION

The techniques of the previous two sections may be generalized to apply to problems of restricted range and derivatives approximation. For ease of notation, we will restrict ourselves to the range of the approximant and its first derivative only. Since the methods of proof are almost totally analogous, we will simply review and state the main results.

As previously, we assume that U c C I (K), dim U = n. Here we are given four functions bo' to' bl , tl E C(K) , where bo(x) < to(x) and bl (x) < tl (x) for all x E K . Set

M(bo' to; bl , t l ) = {u: uE U, bo:S u:S to' bl :::; u':::; t l }.

We assume in what follows that intM(bo' to; b l , tl ) =f:. 0, i.e., there exists a U E U satisfying bo(x) < u(x) < to(x) and bl (x) < u' (x) < tl (x) for all x E K . Our first result is a generalization of Theorem 5.1.

Theorem 7.1. Let M(bo' to; bl , t l ) be as just described, and dim U = n. Then u* is a best L I (K, Jl)-approximantto f E C(K) from M(bo' to; bl ' t l ) if and only if there exist distinct points

(a) {Xi}:1 ~ Z(u* - bo)' (b) {yJ;=1 ~ Z(to - u*), (c) {zi}~=1 ~ Z(u*' - b l ),

(d) {Wi}:=1 ~ Z(tl - u*') with 0 :S m + r + s + t :S n, positive numbers {ai}:I' {Pi};=I' {YJ~=I and {O"Y= I ' and an h E L 00 (K , Jl) satisfying

(1) Ih(x)1 = 1, all x E K, (2) IK hu dJl + '£:1 aiu(xi) - '£;=1 PiU(Yi) + '£~=I Yiu' (Zi) - '£:=1 O"i U' (Wi) =

O,al/uEU, (3) IKhU-u*)dJl=llf-u*III'

Paralleling Theorem 5.3, we obtain this next result, which characterizes the M(bo' to; b l , t l ) that are unicity sets for CI(K, Jl).

Theorem 7.2. Let M(bo' to; bl' t l ) be as shown before, and dim U = n. M(bo' to; bl ' t l ) is a unicity set for CI (K ,Jl) if and only if there do not exist



UI ' U2 E M(bo, to; bl , tl), UI =I- u2' and

(a) {X):I ~Z(ul-bO)nZ(u2-bo)' (b) {y);=1 ~ Z(to - ul ) n Z(to - u2), (c) {Zi}~=1 ~ Z(u; - bl) n Z(u; - bl ), (d) {Wi}~=1 ~ Z(tl - u;) n Z(tl - u;), (e) nonnegative numbers {Ct i } , {Pi}' {y), {oJ,

where 0 ~ m + r + s + t ~ n, and an h E L 00 (K , f..l) Jar which

(1) Ih(x)1 = 1, all x E K, (2) IK hu df..l + 2::1 Ctiu(xJ - 2:;=1 PiU(Yi) + 2:~=1 Yiu' (Zi) - 2::=1 (Ji U' (Wi) =

0, all U E U, (3) hlu, - u21 E C(K) .

Analogous to Property C we have

Definition 7.1. We say that M(bo, to; bl ' t l ) satisfies Property D (with respect to bo , to' bl , and t l ) if for any u l ' u2 E M(bo, to; bl , t l ), u l =f. u2' with {Xi}' {yJ, {z) and {wJ as in Theorem 7.2, 0 ~ m + r + s + t ~ n, and any h E L 00 (K) satisfying

(a) Ih(x)l= l,all XEK, (b) hlu 1 -- u21 E C(K),

there exists a v E U, v =f. 0, for which

(1) v = 0 a.e. on Z(u l - u2), (2) hv 2:: 0 a.e., (3) (a) v(xJ 2:: 0, i = 1 , ... , m,

(b) v (y i) ~ 0, i = 1 , ... , r , (c) v'(z) 2:: 0, i=I, ... ,s, (d) v'(w)~O, i=I, ... ,t.

Finally we obtain

Theorem 7.3. M(bo, to; b l ' t l ) is a unicity set Jar C I (K , f..l) Jar every f..l E 9 if and only iJ M( bo ' to; bl ' t I) satisfies Property D (with respect to bo' to' bl , and t l ).

We note, without proof, that both algebraic polynomials and splines of de-gree at least three with simple, fixed knots are such that M(bo' to; bl ' t l ) sat-isfies Property D on an interval [c, d] for all bo , to' b l , tl E C I [c, d], if intM(bo, to; bl , t l ) =f. 0.

Acknowledgment. The authors wish to thank Professor A. Kroo for various help-ful comments.


L1-APPROXIMATJON WITH CONSTRAINTS 261

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DEPARTMENT OF MATHEMATICS, TECHNION, I.I.T., HAIFA, ISRAEL

INSTITUT FUR ANGEWANDTE MATHEMATIK, UNIVERSITAT ERLANGEN, 8520 ERLANGEN, WEST GERMANY


2....u is a unicity space for c1 (k , f.1) for all f.1 e g if and only if u satisfies property a....

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